The present invention relates to data processing and visualization. More particularly, it relates to a similarity transformation method of representing physical or computer generated data or functions on a discrete grid of one or more independent coordinates for use in a variety of computer applications. In this regard, this method extracts generic shape information on functions of one or more variables, and provides the means to manipulate a function describing a physical system while maintaining the generic shape of the function for a variety of computer applications, such as, for example, function fitting, inversion of data, graphical display and data visualization, pattern recognition, and data synthesis.
Applications that extract generic shape information involve the construction of a parametric representation of the data or object of interest, and then manipulating the values of the parameters to cover the range of states that may be realized by the physical or graphical system of interest.
For example, upper atmospheric remote sensing techniques often measure geophysical properties indirectly, requiring that the underlying variable of interest (e.g., species density) be inferred from the data through comparison with a forward model of measurement process. In discrete inverse theory (DIT), the forward model includes a parametric representation of the variable to be retrieved. The data then provides a basis for computing optimal values of the model parameters.
Consider the remote measurement of altitude profiles of upper atmospheric properties (e.g., species densities or temperatures). Measurement techniques include computerized ionospheric tomography and remote sensing of thermospheric and ionospheric composition using ultraviolet limb-scanning or limb-imaging. In the inversion process, one may parameterize the species altitude profile by one of various means, which include: (1) using an analytic function that is perceived to approximate the xe2x80x9ctruexe2x80x9d function; (2) by identifying model parameters with species concentration values on a discrete vertical grid; and (3) through an expansion in a set of basis functions (i.e., splines or empirical orthogonal functions), which are often truncated to increase computational speed.
In order to manipulate a function governing a physical system, while maintaining generic shape of a function, for achieving function fitting, inversion of data, or pattern recognition, construction of a parametric xe2x80x9cforward modelxe2x80x9d of the measurement process may be needed to compute the optimal values of the parameters by systematic comparison of the forward model values with the measured data. The similarity transformation method of the present invention works well with standard algorithms for computing optimal values.
The task of achieving function fitting, inversion of data, and pattern recognition requires the selected parametric representation to be robust in order to access the range of values that a physical system can occupy. The parametric representation must also be constrained to prevent unrealistic or nonphysical states/values to be accessed through manipulation of the parameters. For example, if one uses an overly robust function to attempt a smoothing of noisy data, the function may xe2x80x9cfit the noisexe2x80x9d rather than the smooth representation desired.
The analytic function approach, as described above, sufficiently constraint the forward model to prevent undue influence by noise. The analytic function approach often requires a minimal number of model parameters to be evaluated. This approach, however, lacks the robustness to capture faithfully all of the possible states of the system or object of interest.
The second and third approaches, as noted above, identify model parameters with species concentration values on a discrete vertical grid, or with coefficients of an expansion in a set of basis functions, respectively, require the evaluation of more model parameters. Further, some form of regularization or a priori information is necessary to ensure smoothness of the retrieved representation in the presence of noise, in order to prevent the models from becoming sufficiently flexible to xe2x80x9cfit the noisexe2x80x9d, or to become computationally unstable. Thus, there is a need for a method to overcome the problems as identified above.
Accordingly, the present invention proposes a method to overcome the above identified problems. The present invention embeds detailed information on the shape of a physical function in a discrete (grid-based) representation. The present method includes advantages of the analytic function approach without the drawback of having to identify or concoct an analytic representation that is both physically faithful and robust. Detailed shape information may be obtained from past discrete data on the system or function of interest, fields of discrete function values derived from detailed simulations or from analytic theory. The similarity transform method of the present invention enables the determination of universality of function shapes in various models or data sets as functions of environmental conditions, location, time, etc. For example, given a species number density profile that is known or assumed to be typical, the similarity transformation method of the present invention produces a parametric function that ranges over the infinite set of profiles having the same generic shape properties (ordering of local extrema, inflection, points, etc.). This explicit shape constraint ensures smoothness in fitting noisy data by the parameterized function.
The present method provides a framework for extracting generic profile shape information, in the form of a non-dimensional shape function, from observations, physics-based numerical simulations, or analytic theory. In this way, the present method facilitates analysis of general characteristics of species concentration variations with coordinates and with other indexing parameters. For DIT retrievals of species concentration profiles from atmospheric observations, the similarity transform-based forward model embeds the generic (xe2x80x9cbasisxe2x80x9d) shape information directly into a parametric representation of each species profile. The representation may also be used to cover the extraction of non-dimensional shape functions from discrete data or simulations, the basic forward model representation, and generalizations of the basic approach.
In another embodiment, the method of the present invention may be used to represent multivariate functions, as well as single variable functions. For multivariate functions, the method involves division of the basis shape function into contiguous hyper-subsurfaces by partitioning the basis shape function domain into contiguous subsets. Likewise, the forward model domain is also partitioned and mapped with the basis function subsurfaces for corresponding subsets of the forward model domain.
In one aspect, a method of providing parameterized representation of geophysical functions for use in retrieving the geophysical functions from remote sensing data, comprising: obtaining atmospheric measurements; extracting generic profile shape information from the measurements; embedding the profile shape information in a parametric discrete grid-based profile representation model (forward model); and retrieving species concentration profiles from the forward model. The data is preferably obtained by remote sensing systems. The data may also be obtained by numerical simulations. The profile shape information is preferably extracted at every latitude-longitude grid point for maintaining an approximate universality of species profile shape under specific geophysical conditions. The shape information is extracted using Discrete Inverse Theory (DIT). The forward model provides a parameterized representation of a signal without statistical noise (true signal). The values of the forward model are manipulated to fit said forward model to said true signal. The method of providing parameterized representation, as above, is performed to accomplish at least one of function fitting, inversion of data, graphical display and data visualization, pattern recognition, or data synthesis functions.
In another aspect, method for extracting generic shape information on functions having one or more variables, comprising: measuring atmospheric data by remote sensing; defining dimensionless similarity variable and dimensionless shape function; extracting discrete values of the shape function; performing function manipulation and retrieval from the extracted discrete values; selecting a basis function; defining a forward model Nf for a ground truth function, the forward model representing an exact profile of the property of interest that underlies the data; performing fitting process on the forward model while maintaining underlying shape function constant; ensuring that the forward model function is similar in shape to the basis function; iterating the step of performing the fitting process if the forward model function is dissimilar in shape to the basis function; and mapping the basis function profile to the forward model.
In yet another aspect, a similarity transform method of providing parameterized representation of geophysical functions for use in retrieving the geophysical functions from remote sensing data, comprising: obtaining function samples; extracting generic profile shape information from the samples; embedding the profile shape information in a parametric discrete grid-based profile representation model (forward model); fitting the forward model to the samples to obtain fitted forward model; and retrieving species concentration profiles from the fitted forward model to retrieve geophysical functions.
Still other objects and advantages of the present invention will become apparent to those skilled in the art from the following detailed description, wherein only the preferred embodiment of the invention is shown and described, simply by way of illustration of the best mode contemplated of carrying out the invention. As will be realized, the invention is capable of different embodiments, and its several details are capable of modifications in various obvious respects, all without departing from the invention. Accordingly, the drawings and description are to be regarded as illustrative in nature, and not as restrictive.